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Papers by Jan Wehr

Physical Review Letters, Apr 28, 2010

We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces act... more We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. We quantify this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces. Our results demonstrate that the force measurement process is prone to artifacts if the noise is not correctly taken into account.

Journal of Statistical Physics, Mar 25, 2015

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schrödinge... more This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schrödinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite.

Journal of Statistical Physics, Apr 15, 2016

We consider the energy difference restricted to a finite volume for certain pairs of incongruent ... more We consider the energy difference restricted to a finite volume for certain pairs of incongruent ground states (if they exist) in the d-dimensional Edwards-Anderson (EA) Ising spin glass at zero temperature. We prove that the variance of this quantity with respect to the couplings grows at least proportionally to the volume in any d ≥ 2. An essential aspect of our result is the use of the excitation metastate. As an illustration of potential applications, we use this result to restrict the possible structure of spin glass ground states in two dimensions.

WORLD SCIENTIFIC eBooks, Mar 1, 2002

Journal of Statistical Physics, Feb 4, 2019

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored ... more We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein-Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

Journal of Statistical Physics, Nov 27, 2018

We study a class of systems whose dynamics are described by generalized Langevin equations with s... more We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in [1], whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath.

Journal of Statistical Physics, Mar 1, 1997

We study sequences of random variables obtained by iterative procedures, which can be thought of ... more We study sequences of random variables obtained by iterative procedures, which can be thought of as nonlinear generalizations of the arithmetic mean. We prove a strong law of large numbers for a class of such iterations. This gives rise to the concept of generalized expected value of a random variable, for which we prove an analog of the classical Jensen inequality. We give several applications to models arising in mathematical physics and other areas.

Physical review, Nov 16, 2018

The presence of a delay between sensing and reacting to a signal can determine the long-term beha... more The presence of a delay between sensing and reacting to a signal can determine the long-term behavior of autonomous agents whose motion is intrinsically noisy. In a previous work [M. Mijalkov, A. McDaniel, J. Wehr, and G. Volpe, Phys. Rev. X 6, 011008 (2016)], we have shown that sensorial delay can alter the drift and the position probability distribution of an autonomous agent whose speed depends on the illumination intensity it measures. In this work, we consider an agent whose speed and rotational diffusion both depend on the illumination intensity and are subject to two independent sensorial delays. Using theory, simulations and experiments with a phototactic robot, we study the dependence of the drift, and of the probability distribution of the robot's position on the sensorial delays. In particular, the radial drift may have positive as well as negative sign, and the position probability density peaks in different regions, depending on the choice of the model's parameters. This not only generalizes previous work, but also explores new phenomena, resulting from the interaction between the two delay variables.

PhDT, 1989

Rigorous results concerning mathematical models of disordered systems in classical statistical me... more Rigorous results concerning mathematical models of disordered systems in classical statistical mechanics are presented. They fall into two categories. Results of the first category are general and dimension-independent. They concern the order of magnitude of fluctuations of extensive quantities; the notion of extensive quantity is introduced in Chapter II and plays a central role there. Results of the second category concern absence of certain phase transitions, caused by randomness (the rounding effect). They are more special and apply only in dimensions which are low enough. The proofs are based on probabilistic tools, such as martingales and moment generating functions.

Communications in Mathematical Physics, Feb 18, 2014

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidea... more We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.

Physical Review Letters, Aug 9, 2011

ABSTRACT A Reply to the Comment by R. Mannella and P. V. E. McClintock.

Physical Review E, Apr 15, 2011

The study of microsystems and the development of nanotechnologies require new techniques to measu... more The study of microsystems and the development of nanotechnologies require new techniques to measure piconewton and femtonewton forces at microscopic and nanoscopic scales. Amongst the challenges, there is the need to deal with the ineluctable thermal noise, which, in the typical experimental situation of a spatial diffusion gradient, causes a spurious drift. This leads to a correction term when forces are estimated from drift measurements [Phys. Rev. Lett. 104, 170602 (2010)]. Here, we provide a systematic study of such effect comparing the forces acting on various Brownian particles derived from equilibrium distribution and drift measurements. We discuss the physical origin of the correction term, its dependence on wall distance, particle radius, and its relation to the convention used to solve the respective stochastic integrals. Such correction term becomes more significant for smaller particles and is predicted to be in the order of several piconewtons for particles the size of a biomolecule.

Reports on Progress in Physics, Apr 18, 2016

Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling o... more Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales become very small. In the presence of multiplicative noise (i.e. noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift or effective drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: the Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient; the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback; and the behavior of an autonomous agent subject to sensorial delay in a noisy environment. This allows us to present the experimental results, as well as mathematical methods and numerical techniques, that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment.

Communications in Mathematical Physics, Nov 27, 2014

We study a class of systems of stochastic differential equations describing diffusive phenomena. ... more We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.

Journal of Mathematical Physics, May 1, 2010

Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising... more Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in R d. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.

Journal of physics, Dec 21, 1995

We show that the distribution of the percolation threshold in a large finite system does not conv... more We show that the distribution of the percolation threshold in a large finite system does not converge to a Gaussian when the size of the system goes to infinity, provided that the two widely accepted definitions of correlation length are equivalent. The shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length. The result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v. All results are rigorously proven in the 2D case. Generalizations for three dimensions are also discussed.

Journal of Statistical Physics, Aug 1, 1990

An extensive quantity is a family of functions gZv of random parameters, indexed by the finite re... more An extensive quantity is a family of functions gZv of random parameters, indexed by the finite regions V (subsets of yd) over which gZv are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lira 7*v/[ V[ has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume of V. Of particular interest is the free energy of a system with random couplings; for such 7* v bounds are derived also for the generating function E(e'V). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.

Springer eBooks, 1992

Condensed matter physics often has to consider systems with static disorder [6], [7], i.e. with i... more Condensed matter physics often has to consider systems with static disorder [6], [7], i.e. with impurities, dislocations, substitutions etc. which vary from sample to sample (thus introducing disorder) but which do not exhibit thermal fluctuations on relevant time scales (hence the word static). To account for such disorder mathematically one often uses lattice spin systems with random parameters in the interaction (e.g. random magnetic fields or random coupling constants). For each fixed realization of these parameters one then obtains a spin system in which the usual quantities of physical interest — magnetization, free energy etc. — can be calculated. Random parameters of this type are often called quenched, to stress the fact that they remain constant during the calculation of spin averages — corresponding to the static nature of the disorder in the modelled physical system.

Journal of Statistical Physics, Nov 5, 2015

We study ground states of Ising models with random ferromagnetic couplings, proving the trivialit... more We study ground states of Ising models with random ferromagnetic couplings, proving the triviality of all zero-temperature metastates. This unexpected result sheds a new light on the properties of these systems, putting strong restrictions on their possible ground state structure. Open problems related to existence of interface-supporting ground states are stated and an interpretation of the main result in terms of first-passage and random surface models in a random environment is presented. 1. Ground states of disordered Ising ferromagnets Consider a system of (classical) Ising spins σ j on the lattice Z d , interacting by a random Hamiltonian H J (σ) = − |i−j|=1 J ij σ i σ j. (1) Here J = {J ij : i, j ∈ Z d , |i − j| = 1} is a realization of an IID family of random variables J ij , which are assumed continuously distributed (i.e. without atoms) and positive (ferromagnetic) and σ is a configuration